UvA-DARE (Digital Academic Repository) An infinite-dimensional affine stochastic volatility model

We introduce a flexible and tractable infinite-dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein– Uhlenbeck-type process, whose instantaneous covariance is given by a pure-jump stochastic process taking values in the cone of positive self-adjoint Hilbert– Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine , that is, we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalized Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state-dependent jump intensity. Infinite dimensional volatility models arise, for example, when considering the dynamics of forward rate functions in the Heath–Jarrow–Morton– Musiela (HJMM) modeling framework using the Filipović space. In this setting, we discuss various examples: an infinite-dimensional version of the Barndorf–Nielsen–Shephard stochastic volatility model, as well as covariance processes with a state dependent intensity.


K E Y W O R D S
forward price dynamics, Heath-Jarrow-Morton-Musiela framework, infinite-dimensional affine processes, Riccati equations, state-dependent jump intensity, stochastic volatility

INTRODUCTION
In this paper, we propose a new class of affine stochastic volatility models (  ,   ) ≥0 , where (  ) ≥0 takes values in a real separable Hilbert space (, ⟨⋅, ⋅⟩  ) and (  ) ≥0 is a timehomogeneous affine Markov process taking values in  + =  + 2 (), the cone of positive selfadjoint Hilbert-Schmidt operators on .The process  is taken from a class of affine processes introduced in Cox et al. (2020).The process (  ) ≥0 is modeled by the following stochastic differential equation: where  ∶ dom() ⊆  →  is a possibly unbounded operator with dense domain dom(), and (   ) ≥0 is a -Brownian motion independent of , with  a positive self-adjoint trace-class operator on .Assuming that  is progressively measurable and using moment bounds on  established in Cox et al. (2020), the existence of a solution to Equation (1) is straightforward (see Lemma 2.8 below).
In Section 2.1, we show that under the assumption that the Markov process (  ) ≥0 has càdlàg paths, it is a square-integrable semimartingale.This follows from the formulation of an associated martingale problem in terms of what we call a weak generator (see Definition 2.2) of the Markov process (  ) ≥0 and yields the explicit representation of (  ) ≥0 as (  , d) where ,  ∈  + ,  ∈ () is a bounded linear operator, given  ∈  + the measure (, ⋅) ∶ ( + ⧵ {0}) → ℝ is such that   (d, d) = (  , d)d is the predictable compensator of the jump-measure of (  ) ≥0 , and (  ) ≥0 is a purely discontinuous  + -valued square integrable martingale.Moreover, by exploiting the results in Cox et al. (2020) and Metivier (1982), we adapt the proof of (Jacod & Shiryaev, 2003, Theorem II.2.42) to our infinite-dimensional setting to obtain the characteristic triplet (see Definition 1.1) of (  ) ≥0 explicitly and show its affine form (see Proposition 2.6).The detailed parameter specifications are given in Assumption 2.1 below.
Our main motivation for studying Hilbert space-valued stochastic volatility models is the modeling of forward prices in commodity or fixed-income markets under the Heath-Jarrow-Morton-Musiela (HJMM) modeling paradigm (see for example, Benth andKrühner (2014, 2015), Filipović (2001), Carmona and Tehranchi (2006), Cont (2005)).In finite dimensions, multivariate stochastic volatility models with state-dependent volatility dynamics driven by Brownian noise and jumps are considered for example in Gourieroux and Sufana (2010), Caversaccio (2014), Leippold and Trojani (2008).The variance process  that we consider generalizes the Lévy-driven case considered in Benth et al. (2018) to a model allowing for state-dependent jump intensities, while 14679965, 2022, 3, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mafi.12347 by Uva Universiteitsbibliotheek, Wiley Online Library on [22/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License maintaining the desired affine property, which makes these models tractable.Stochastic volatilities with jumps describe the financial time series in energy and fixed-income markets well, as it is illustrated, for example, in Eydeland and Wolyniec (2002), Benth and Šaltyt ė Benth (2012), Cont (2001), Leippold and Trojani (2008).We refer in particular to Leippold and Trojani (2008) in which the authors discussed convincing empirical evidence for state-dependent jumps in the volatility.
The proof Theorem 3.3, that is, of the affine property of our stochastic volatility model (  ,   ) ≥0 , is in Section 3. It involves considering an approximation ( ()  ,   ) ≥0 of (  ,   ) ≥0 obtained by replacing  in Equation ( 1) by its Yosida approximation.The use of the approximation allows us to exploit the semimartingale theory and standard techniques in order to show that the approximating process is affine.To show that the affine property holds for the limiting process (  ,   ) ≥0 , we study the convergence of the generalized Riccati equations associated with ( ()  ,   ) ≥0 to those associated with (  ,   ) ≥0 .We prove the existence of a unique solution to these generalized Ricatti equations by exploiting infinite dimensional ODE results and using the quasi-monotonicity argument to show that the solution stays in the cone  + , see Deimling (1977) and Martin (1976).In order for the approach described above to succeed, we impose a commutativity-type condition on the covariance operator of the -Wiener process (   ) ≥0 and the stochastic volatility ( 1∕2  ) ≥0 (see Assumption 2.11 below).This condition is also imposed in Benth et al. (2018) and is rather limiting.However, we show that it can be avoided by considering a slightly different stochastic volatility model, see Remark 2.12 and the example in Section 4.4.
In Section 4, we consider a number of examples.For the process , we assume the setting proposed in Filipović (2001), Benth and Krühner (2014), which can be used to model arbitrage-free forward prices at time  ≥ 0 of a contract delivering an asset (commodity) or a stock at time  + .In this case, the operator  in Equation ( 1) is given by  = ∕ and the space  is given by a Filipović space.For the process (  ) ≥0 , we construct several examples in which we specify the drift and the jump parameters.We first show that the infinite dimensional lift of the multivariate Barndorf-Nielsen-Shephard model introduced in Benth et al. (2018) is a particular example of our model class.The stochastic variance process (  ) ≥0 in this example is a stochastic differential equation driven by a Lévy subordinator in the space of self-adjoint Hilbert-Schmidt operators, as we show in Section 4.1.1.As mentioned above, this example does not involve state-dependent jump intensities.However, Sections 4.2-4.4provide explicit paramater choices that do involve state-dependent jump intensities.In Section 4.2, we construct a variance process, which is essentially one-dimensional; evolving along a fixed vector  ∈  + .In Section 4.3, we construct a truly infinite-dimensional variance process .In this example, both   ,  ≥ 0, and  share a fixed orthonormal basis of eigenvectors.This is imposed to ensure that the commutativity condition given by Assumption 2.11 is satisfied.In Section 4.4, we avoid this commutativity condition by considering an example involving the alternative model discussed in Remark 2.12.In a subsequent article, we plan to compute option prices on forwards in commodity markets based on the models introduced here.In practice, these computations require the study of finite dimensional approximations of the variance process and its associated Ricatti equations, which is being tackled in the working paper Karbach (2022).

Layout of the article
In Section 2, we give an in-depth analysis of our stochastic volatility model and introduce sufficient parameter assumptions that ensure the well-posedness of our proposed model.Subsequently, in Section 3, we prove the affine-property of our joint model (  ,   ) ≥0 .We split the proof into two parts, first in Section 3.1, we show the existence and uniqueness of solutions to the associated generalized Riccati equations under admissible parameter assumptions, thereafter in Section 3.2, we prove the affine transform formula.In Section 4, we give several examples of stochastic volatility models included in our model class by specifying various variance processes (  ) ≥0 .

Notation
For (, ), a topological vector space and  ⊂  we let () denote the Borel--algebra generated by the relative topology on .We denote by   ([0, ]; ) the space of -valued -times continuously differentiable functions on [0, ].
Two -valued locally square-integrable martingales  and  are called orthogonal if the realvalued process (⟨  ,   ⟩) ≥0 is a local martingale.Further, we call  a purely discontinuous local martingale if it is orthogonal to all continuous local martingales.An -valued semimartingale can be written as (see (Metivier, 1982, Theorem 20.2)) where  0 is  0 -measurable,   is a continuous local martingale with   0 = 0,   is a locally square integrable martingale orthogonal to   with   0 = 0, and  is a càdlàg process of finite variation with  0 = 0.The process   in Equation ( 8) is unique (up to a ℙ null set), see (Metivier, 1982, Chapter 4, Exercise 13).We associate with the -valued semimartingale , the integer-valued random measure where   denotes the Dirac measure at point .Recall from (Jacod & Shiryaev, 2003, Theorem II.1.8),the existence and uniqueness (up to a ℙ-null set) of the predictable compensator   of   .Given a semimartingale , we define the "large jumps" process X by and we define the "small jumps" process Since ‖Δ X‖ ≤ 1, X is a special semimartingale and hence it admits the unique decomposition where  0 is  0 -measurable,  X is a local martingale with  X 0 = 0, and  X is a predictable process of finite variation with  X 0 = 0. We are ready to introduce the characteristic triplet of an -valued semimartingale .Definition 1.1.Let  be an -valued semimartingale, let  X be the predictable process of finite variation from decomposition (12), let   be the continuous martingale part of  as provided by Equation (8), and let   be the predictable compensator of   , where   is defined by Equation (9).Then we call the triplet ( X , ⟨⟨  ⟩⟩,   ) the characteristic triplet of .Note that the characteristic triplet consists of a predictable càdlàg -valued process of finite variation, a predictable càdlàg  1 ()-valued process of finite variation, and a predictable random measure on ([0, ∞) × ).

THE STOCHASTIC VOLATILITY MODEL
In this section, we specify our stochastic volatility model.First, in Section 2.1, we introduce the stochastic variance process , which is an affine Markov process on the cone of positive self-adjoint Hilbert-Schmidt operators, the existence of which is established in Cox et al. (2020).We show that whenever the process  admits for a version with càdlàg paths, this version is actually a Markov semimartingale with characteristic triplet of an affine form and the representation (2) holds true.Subsequently, in Section 2.2, we show that given such a stochastic variance process  there exists a mild solution  to Equation (1) with initial value  ∈ , which enables us to introduce our joint stochastic volatility model  = (, ) (see Definition 2.9 below).
Due to the lack of local compactness of the underlying state space, standard Feller theory cannot be employed to establish Theorem 2.3.We overcame this problem by using generalized Feller semigroups, see (Cox et al., 2020, Section 4) and Cuchiero and Teichmann (2020).Unfortunately, the Markov processes associated to a generalized Feller semigroup need not have càdlàg paths (but see (Cuchiero & Teichmann, 2020, Theorem 2.13) for a positive result).Some (rather limiting) conditions that ensure that Assumption 2.4 is satisfied are provided in the lemma below.In ongoing work Karbach (2022), we hope to establish that in fact, Assumption 2.4 is always satisfied.
Lemma 2.5.Assume that (, , , ) is an admissible parameter set that fulfills either one of the following two cases: Then the affine Markov process (  ) ≥0 associated to (, , , ) admits for a version with càdlàg paths.Proof.To prove (i), observe that the weak generator (18) associated to the admissible parameters (, , , 0) is a weak generator of a Lévy driven SDE as described for example in (Peszat & Zabczyk, 14679965, 2022, 3, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mafi.12347 by Uva Universiteitsbibliotheek, Wiley Online Library on [22/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 2007, equation 9.37)) and hence the assertion follows from (Peszat & Zabczyk, 2007, Theorem 4.3).In case of (ii), the assertion follows from (Cox et al., 2020, Proposition 4.19).□ We show in the next proposition that the version of  with càdlàg paths is in fact a Markovian semimartingale.
Proposition 2.6.Suppose that (, , , ) is an admissible parameter set according to Assumption 2.1 and such that the associated affine Markov process  satisfies Assumption 2.4.Then there exists a version of (  ) ≥0 , which is an  + -valued semimartingale with semimartingale characteristics (, ,   ) of the form Moreover, the following representation holds where  is a purely discontinuous square-integrable martingale.
In order to prove Proposition 2.6, we need the following result, which can be obtained by mimicking the proof of (Peszat & Zabczyk, 2007, Proposition 9.38).

The joint stochastic volatility model
In this section, we present our joint model, see Definition 2.9 below, which involves taking the square root  1∕2 of the process  from Theorem 2.3 as volatility for the -valued process  given by Equation ( 29) below.
Throughout this section, we consider the following setting: let (, , , ) be a parameter set satisfying Assumption 2.1, let  ∈  + and  ∈ , and let  ∈  1 () be self-adjoint and positive.Next, let  be the square-integrable time-homogeneous Markov process associated with the parameter set (, , , ) the existence of which is guaranteed by Theorem 2.3; we denote the filtered probability space on which  is defined by (Ω 1 ,  1 , ( 1  ) ≥0 , ℙ 1 ) and assume ℙ 1 ( 0 = ) = 1.In addition, we let (Ω 2 ,  2 , ( 2  ) ≥0 , ℙ 2 ) be another filtered probability space, which satisfies the usual conditions and allows for a -Wiener process and denote the expectation with respect to ℙ by .With slight abuse of notation, we consider  and   to be processes on (Ω,  , ) (note that they are independent).In addition, we assume (, dom()) to be the generator of a strongly continuous semigroup (()) ≥0 on .
Now consider the following SDE, for which Lemma 2.8 below establishes the existence of a mild solution Lemma 2.8.Assume the setting described above, in particular, let (, , , ) satisfy Assumption 2.1 and let  be the associated affine process.Moreover, let Assumption 2.4 hold.Then  is progressive, and moreover is the unique mild solution to Equation ( 29).
Remark 2.10.The assumption that   is a -Wiener .processcan be weakened whilst maintaining all results presented in this article.Indeed, as  itself is already  + valued, it suffices to assume that  ∈  2 () (instead of  ∈  1 ()) (see also the proof of Lemma 2.8).
In order to show that our joint model is affine (see Theorem 3.3 below), we need one further assumption.This assumption is also imposed in Benth et al. (2018) To the best of our knowledge, all examples for which Assumption 2.11 holds are such that  and   commute for all  ≥ 0. In fact, as commuting self-adjoint and compact operators are jointly diagonizable, this is difficult to ensure without assuming there exists a fixed orthonormal basis (  ) ∈ℕ of  that forms the eigenvectors of  and of   ,  ≥ 0. Note that this essentially reduces the state space of  to the cone of positive, square-integrable sequences  + 2 , that is, we only model the eigenvalues of , as the eigenvectors are fixed, see also Section 4.3.In conclusion, Assumption 2.11 is rather limiting.However, it can be circumvented if one considers a slightly different model, see Remarks 2.12 and 2.13 below.
Remark 2.12.Assumption 2.11 can be omitted if, instead of Equation ( 29), one assumes that the process  in the joint model satisfies the following stochastic differential equation: where  is an -cylindrical Brownian motion (i.e., d  is white noise) and  ∈  1 () is positive and self-adjoint (in fact,  ∈  + suffices, see Remark 2.10).In this case, provided Assumptions 2.1 and 2.4 hold, we have and is the unique mild solution to Equation (38), see also (Da Prato & Zabczyk, 1992, Chapter 4, Section 3).Moreover, Theorem 3.3 below remains valid: if  is given by Equation ( 38) and Assumptions 2.1 and 2.4 hold, we obtain exactly the same expression for [e ⟨  , 1 ⟩  −⟨  , 2 ⟩ ].In particular the joint model involving Equation (38) under Assumptions 2.1 and 2.4 coincides with the joint model involving Equation (29) under Assumptions 2.1-2.11, in the sense that for every fixed time  ≥ 0, the distribution of (  ,   ) is the same.We refer to Section 4.4 for an example of a joint model involving Equation (38).
Remark 2.13.If (, dom()) is the generator of an analytic semigroup and moreover ), then a mild solution to Equation (29) exists even if   is an -cylindrical Brownian motion.These conditions are satisfied, for example, when  is the Laplacian on ℝ  for  ∈ {1, 2, 3}.We refer to Da Prato and Zabczyk (1992) for details.
Although this provides another way to circumvent Assumption 2.11 (as  is the identity in this case), we will not investigate this setting any further: for the applications we have in mind (, dom()) fails to be the generator of an analytic semigroup.Note that to obtain the assertions of Theorem 3.3 in this setting, one would have to adapt its proof: one would not only have to approximate the operator  but also the noise.

THE JOINT STOCHASTIC VOLATILITY MODEL IS AFFINE
In this section, we present our main result, namely that the stochastic volatility model  = (, ) in Definition 2.9 has the affine property, see Theorem 3.3.In particular, this means that we can express the mixed Fourier-Laplace transform [e ⟨  ,⟩  −⟨  ,⟩ ] ( ∈ ,  ∈  + ) in terms of the solution to generalized Riccati equations associated to the model parameters (, , , ),  and  (respectively ).In the upcoming subsection, we discuss the well-posedness of these generalized Ricatti equations.Our main result, Theorem 3.3, is contained and proven in Section 3.2.

The affine property of our joint stochastic volatility model
Exploiting the existence of a solution to the generalized Riccati equation ( 43), we show in the following theorem that our joint stochastic volatility model  = (, ) in Definition 2.9 has indeed the affine property.43), the existence of which is guaranteed by Proposition 3.2.Then for all  ≥ 0, it holds that  [ e ⟨  , 1 ⟩  −⟨  , 2 ⟩ ] = e −Φ(,)+⟨, 1 (,)⟩  −⟨, 2 (,)⟩ . (55) In applications, we are usually interested in distributional properties of the process (  ) ≥0 .Setting  2 = 0 in Equation ( 55), we obtain a quasi-explicit formula for the characteristic function of   for  ≥ 0. Due to its importance, we state it as a (trivial) corollary of Proposition 3.3.

EXAMPLES
In this section, we discuss several examples that are included in our class of joint stochastic volatility models with affine pure-jump variance.In all the examples, we assume that the first component  is modeled in the abstract setting of Definition 2.9, that means we do not specify  or  any further, however, we stress here that the HJMM modeling framework as described in Filipović ( 2001), Benth and Krühner (2014) ) is an increasing continuous function such that  −1∕3 is integrable, for example, () = e  for some constant  > 0. Then  is indeed a separable Hilbert space when equipped with the inner product ⟨, ⟩  = (0)(0) + ∫ ∞ 0 () ′ () ′ () d.In this setting, the operator  is given as the first derivative in space, that is,  = ∕.Our focus here is on correct specifications of the parameter set (, , , ) and the initial value  0 =  ∈  + such that Assumption 2.1 holds and the associated process (  ) ≥0 satisfies Assumption 2.4 as well as the joint process (, ) satisfies Assumption 2.11.
In Section 4.1, we show that an Ornstein-Uhlenbeck process driven by a Lévy subordinator in  + is included in our model class for the variance process , which is implied by the parameter choice  = 0. Consequently, in Section 4.1.1,we conclude that our class of stochastic volatility models extends the infinite-dimensional lift of the BNS stochastic volatility model introduced in Benth et al. (2018).
In the subsequent examples, we focus on variance processes admitting for state-dependent jump intensities.Comparable to the Lévy-driven case, these examples have the advantage to model the volatility clustering phenomenon.This was, for example, illustrated in Leippold and Trojani (2008) in a finite-dimensional setting where the state space is the cone of symmetric positive semi-definite matrices.In this latter paper, it was shown in a numerical example that for this type of models, the volatilities and jump intensities are time-varying leading to a clustering of jump events in phases of high jump intensities.
In Section 4.2, we construct a variance process , which takes values in { +  ∶  ≥ 0} for some fixed  ∈  + .This is somewhat of a toy model: although the variance process is infinitedimensional, its randomness is one-dimensional.In Section 4.3, we consider a truly infinitedimensional stochastic variance process .However, to ensure that Assumption 2.11 is satisfied, we assume that both  and   ,  ≥ 0 are diagonizable with respect to the same fixed orthonormal basis.We close this section with Section 4.4 in which we show the benefits of the model discussed in Remark 2.12, which does not require Assumption 2.11 and thus allows for a more general variance process.

The operator-valued BNS SV model
In Benth et al. (2018), the authors introduced an operator-valued volatility model that is an extension of the finite-dimensional model introduced in Barndorff-Nielsen and Stelzer (2007) (and thus they named it the operator-valued BNS SV model).In their model, it is assumed that the volatility process  is driven by a Lévy process (  ) ≥0 .In order to ensure that  is positive, they assume that  ↦   is almost surely increasing with respect to  + , that is, that  is an  + -subordinator.This holds if and only if for any fixed  ≥ 0, we have ℙ(  ∈  + ) = 1, (see also (Pérez-Abreu & Rocha-Arteaga, 2006, Proposition 9)).Roughly speaking, the model considered in Benth et al. (2018) amounts to taking  ≡ 0 in our setting (i.e., to considering a stochastic volatility model  = (, ) in Definition 2.9 with parameters (, , , 0, , )).Indeed, in Section 4.1.1below, we demonstrate that the model introduced in Benth et al. ( 2018) is fully contained in our setting.
In this section, we show that the joint volatility model ( 73) is a special case of our model in the case that  ≡ 0, more specifically, that (Benth et al., 2018, Proposition 3.2) is a special case of Proposition 4.1 above.To this end, we first remark that if  ∈  2 (),  ∈  1 ( 2 ()), and  ∶ ( 2 ()) → [0, ∞] are the characteristics of , then |  ≡ 0 thanks to (Benth et al., 2018, Proposition 2.10).Moreover, in view of Lemma A.2, we have that  ∈ ,  = 0, and supp() ⊆  (this answers an open question in Benth et al. (2018): see the discussion prior to Proposition 2.11 in that article).Finally, it is easily verified that () ⊂  in both cases described above, so although the 'ambient' space for  is  2 () in Benth et al. (2018), one can, without loss of generality, take  as ambient space for .

An essentially one-dimensional variance process
We now present a simple example of a pure-jump affine process (  ) ≥0 on  + with statedependent jump intensity.Starting from its initial value  0 =  ∈  + this process moves along a single vector  ∈  + ⧵ {0} and is thus essentially one-dimensional.For this case we specify an admissible parameter set (, , , ) such that the associated affine process  has càdlàg paths and is driven by a pure-jump process (  ) ≥0 with jumps of size  ∈ (0, ∞) in the single direction  ∈  + with ‖‖ = 1 and such that the jump-intensity depends on the current state of the process .For the sake of simplicity, we let the constant parameters  and  be zero.Moreover, we shall fix the dependency structure by means of a fixed vector  ∈  + ⧵ {0}.We then take a measure  ∶ ((0, ∞)) → [0, ∞) such that ∫ ∞ 0  −2 (d) < ∞ and define the vector valued measure From the assumption that ∫ ∞ 0  −2 (d) < ∞ it follows that for every  ∈  + the measure (, d) on ( + ⧵ {0}) defined by is finite and thus also We now must find a linear operator  ∶  →  such that

A state-dependent stochastic volatility model on a fixed ONB
In this example we specify an admissible parameter set (, , , ) giving more general affine dynamics of the associated variance process  on  + .In the previous Section 4.2 we imposed additional commutativity assumptions on the initial value  0 =  ∈  + , the jump direction  and the covariance operator .In this example we allow for a more general jump behavior, while maintaining Assumption 2.11.To do so, we pick up the discussion preceding Remark 2.12 and note here that Assumption 2.11 is satisfied, whenever  and   commute for all  ≥ 0. Recall that  and (  ) ≥0 commute if and only if they are jointly diagonizable.This motivates the consideration of a variance process  that is diagonizable with respect to a fixed ONB.More concretely, let (  ) ∈ℕ be an ONB of eigenvectors of the operator .We model  such that   ( ≥ 0) is diagonizable with respect to the ONB (  ) ∈ℕ , i.e.
for the sequence of eigenvalues (  ()) ∈ℕ of   in  + 2 .Concerning the modeling of the dynamics of (  ) ≥0 , this essentially means that we model the dynamics of the sequence of eigenvalues (  ()) ∈ℕ in  + 2 only.We now come to a specification of the parameters (, , , ) such that Assumption 2.1 is satisfied and moreover such that   is indeed diagonizable with respect to (  ) ∈ℕ for all  ≥ 0. Let the measure  ∶ ( + ⧵ {0}) → [0, ∞) be such that for  ∈ ( + ⧵ {0}) we have for a sequence (  ) ∈ℕ of finite measures on ((0, ∞)) such that for a sequence of finite measures (  ) ∈ℕ on ((0, ∞)) such that Moreover, let  ∈  be diagonizable with respect to (  ) ∈ℕ , note that this implies that for any  ∈  + that is diagonizable with respect to (  ) ∈ℕ , we have that  +  * is diagonizable with respect to (  ) ∈ℕ as well.We thus define the linear operator  ∶  →  by Now, one can check that  and  indeed satisfy their respective conditions in Assumption 2.1.Due to the first condition on  in (81) and the second on  in (84), it follows from Proposition 2.5 that Assumption 2.4 is satisfied.
Again from the semimartingale representation (22) we conclude that for all  ≥ 0 the operator   is diagonizable with respect to (  ) ∈ℕ and thus Assumption 2.11 is satisfied as well.

A general state-dependent stochastic volatility model
In this example we show that modeling under the alternative formulation of the model (, ) provided by Remark 2.12 gives considerably more freedom in the model parameter specification.We write b =  + ∫  + ∩{‖‖>1}  (d) and for every  ∈  we set B() = () + ∫  + ∩{‖‖>1}  ⟨,(d)⟩ ‖‖ 2 .We then see that for the stochastic volatility model (, ) given by the SDE with ( 0 ,  0 ) = (, ) ∈  ×  + , and  = (  ) ≥0 a cylindrical Brownian motion, the Assumption 2.11 can be dropped.Therefore, every admissible parameter set (, , , ), such that the associated affine process  satisfies Assumption 2.4 is a valid parameter choice.To emphasize the gained flexibility, we compare it with the example in Section 4.3.For simplicity, we let (  ) ∈ℕ be some ONB of  and specify  and  as in ( 80) and ( 83), respectively, with respect to this ONB.This means that the noise in the variance process  again occurs on the diagonal only.However,  need not be diagonizable with respect to (  ) ∈ℕ and instead of taking  to be diagonizable with respect to the ONB (  ) ∈ℕ and  of the particular form ( 85 where  is as in Equation ( 21) and  is a purely discontinuous square integrable martingale with a compensator   .The dynamics (89) has a similar structure as the affine dynamics of covariance processes in finite dimensions presented in (Cuchiero et al., 2011, equation 1.2) in the purejump case.Indeed, both models have an affine drift and are driven by a pure-jump process whose compensator is an affine function of .As mentioned above, this model will also demonstrate clustering behavior.

CONCLUSION AND OUTLOOK
In Section 2, we introduce an infinite dimensional stochastic volatility model.More specifically, we consider a process  that solves a linear SDE in a Hilbert space  with additive noise, where the variance of the noise is dictated by a process  taking values in the space of self-adjoint Hilbert-Schmidt operators on .The process  is assumed to be an affine pure-jump process that allows for state-dependent jump intensities; its existence has been established in the previous work Cox et al. (2020) under certain admissibility conditions on the parameters involved (see Assumption 2.1).
In the derivation of the affine transform formula, we make use of Hilbert-valued semimartingale calculus, for this reason, we must assume that  has càdlàg paths (see Assumption 2.4).Currently, we establish existence of càdlàg paths under limited conditions (see Proposition 2.5).Relaxing these conditions is one of the aims of the working paper Karbach (2022) where the author considers finite-dimensional approximations (in particular, Galerkin approximations of the associated generalized Riccati equations are considered) and studies convergence of the variance process in the Skorohod topology.Having introduced the joint model, we prove that it is affine (see Theorem 3.3).To this end, we need an additional "commutativity"-type assumption, see Assumption 2.11.This assumption is avoided by considering a slightly different model, see Remark 2.12 and Section 4.4.
Our model extends the model introduced in Benth et al. (2018), where the authors assume that  is driven by a suitably chosen Lévy process (see Section 4.1.1).In Section 4, we also discuss other concrete examples of our model.
Another way to avoid Assumption 2.11 would be to construct a variance process  that takes values in the space of self-adjoint trace class operators.Indeed, in this case, we can assume that the process   driving  is a cylindrical Brownian motion (i.e.,  is the identity).However, taking the trace class operators as a state space is not trivial as this is a nonreflexive Banach space.We aim to pursue this direction of research in a forthcoming work.
Finally, in a subsequent work, we plan to consider the dynamics of forward rates in commodity markets modeled by our proposed stochastic volatility dynamics.Then study the problem of computing option prices on these forwards.In practice, these computations require finite-rank approximations of the associated generalized Riccati equations as being considered in Karbach (2022).

A C K N O W L E D G M E N T
Corresponding author is funded by The Dutch Research Council (NWO); Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Grant No: C.2327.0099).

D ATA AVA I L A B I L I T Y S TAT E M E N T
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),  ∈ [0, ] is a local martingale.Furthermore, since it is bounded on [0, ], it is a martingale and it holds